Strength versus gravity - UChicago GeoScigeosci.uchicago.edu/~kite/doc/ch3_of_melosh.pdf52 Strength versus gravity 3.2 Stress and strain: a primer a full exposition of the continuum

49

3

Strength versus gravity

the existence of any differences of height on the earth’s surface is

decisive evidence that the internal stress is not hydrostatic. If the earth

was liquid any elevation would spread out horizontally until it disap-

peared. the only departure of the surface from a spherical form would

be the ellipticity; the outer surface would become a level surface, the

ocean would cover it to a uniform depth, and that would be the end of us.

the fact that we are here implies that the stress departs appreciably from

being hydrostatic; …

H. Jeffreys, Earthquakes and Mountains (1935)

3.1 Topography and stress

Sir Harold Jeffreys (1891–1989), one of the leading geophysicists of the early twentieth

century, was fascinated (one might almost say obsessed) with the strength necessary to

support the observed topographic relief on the earth and Moon. through several books and

numerous papers he made quantitative estimates of the strength of the earth’s interior and

compared the results of those estimates to the strength of common rocks.

Jeffreys was not the only earth scientist who grasped the fundamental importance of rock

strength. almost ifty years before Jeffreys, american geologist G. K. Gilbert (1843–1918)

wrote in a similar vein:

If the earth possessed no rigidity, its materials would arrange themselves in accordance with the laws

of hydrostatic equilibrium. the matter speciically heaviest would assume the lowest position, and

there would be a graduation upward to the matter speciically lightest, which would constitute the

entire surface. the surface would be regularly ellipsoidal, and would be completely covered by the

ocean. elevations and depressions, mountains and valleys, continents and ocean basins, are rendered

possible by the property of rigidity.

G. K. Gilbert, Lake Bonneville (1890)

By rigidity Gilbert meant the resistance of an elastic body to a change of shape. He was

well aware that this rigidity has its limits, and that when some threshold is exceeded earth

materials fail to support any further loads. We call this threshold strength and recognize

that this material property resists the tendency of gravitational forces to erase all topo-

graphic variation on the surface of the earth and the other solid planets and moons.

http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/CBO9780511977848.004Downloaded from http:/www.cambridge.org/core. University of Chicago, on 04 Jan 2017 at 02:46:58, subject to the Cambridge Core terms of use, available at

http:/www.cambridge.org/core/termshttp://dx.doi.org/10.1017/CBO9780511977848.004http:/www.cambridge.org/core

Strength versus gravity50

the importance of strength is highlighted by a simple computation that Jeffreys included

in his masterwork, The Earth (1952). this computation is summarized in Box 3.1, where

it is shown that, without strength, a topographic feature of breadth w would disappear from

the surface of a planet in a time tcollapse given by:

tw

gcollapse =

π8

(3.1)

where g is surface gravitational acceleration. Without strength, a mountain 10 km wide on

the earth would collapse in about 20 seconds, and a 100 km wide crater on the moon would

disappear in about 3 minutes. clearly, such features can and do persist for much longer

periods of time.

Planetary topography, and the material strength that makes it possible, lend interest

and variety to planetary surfaces. However, when seen from a distance, it is clear that

the shapes of planets are, nevertheless, very close to spheroids. only very small aster-

oids and moons (Phobos and Deimos are examples) depart greatly from a spheroidal

shape in equilibrium with their rotation or tidal distortion. thus, although the strength of

planetary materials (rock or ice) is adequate to support a certain amount of topography,

it is evidently limited. Such things as 100 km high mountains do not exist on the earth

because strength has limits. the ultimate extremes of altitude on a planet’s surface are

regulated by the antagonism between the strength of its surface materials and its gravi-

tational ield.

although everyone has an intuitive idea of strength, the full quantiication of this property

is both complex and subtle. Many introductory physics or engineering textbooks present

strength as if it were a simple number that can be looked up in the appropriate handbook.

this impression is reinforced by handbooks that offer tables of numbers purporting to

represent the strength of given materials. But further investigation soon reveals that there

are different kinds of strength: crushing strength, tensile strength, shear strength, and many

others. Strength sometimes seems to depend on the way that forces or loads are applied to

the material, and upon other conditions such as pressure, temperature, and even its history

of deformation. the various strengths of ductile metals, like iron or aluminum, typically do

not depend much on how the load is applied, or how fast it is applied, but common planet-

ary materials behave quite differently.

Quantitative understanding of the relation between topography, strength, and gravity

requires, irst, some elementary notions of stress and strain and, second, a more detailed

understanding of how apparently solid materials resist changes in shape. this chapter intro-

duces the basic concepts of stress, strain, and strength before failure, and applies them to

the limits on possible topography. It also introduces the role of time and temperature in

limiting the strength of materials and the duration of topographic features. the next chapter

examines deformation beyond the strength limit and the tectonic landforms that develop

when this limit is exceeded.



3.1 Topography and stress 51

Box 3.1 Collapse of topography on a strengthless planet

consider a long mountain ridge of height h, width w and effectively ininite length L standing

on a wide, level plain. for simplicity suppose that the proile of the mountain is rectangular,

with vertical cliffs of height h bounding both sides (figure B3.1.1). the surface gravitational

acceleration of the planet on which this mountain lies is g, and ρ is the density of the material from which both the mountain and planetary surface are composed.

the weight of the mountain is ρghwL. If there is no strength, this weight (force) can only be balanced by the inertial resistance of material accelerating beneath the surface, according to

newton’s law F = ma. the driving force F equals the weight of the mountain, F = ρghwL. the

acceleration a is equal to the second time derivative of the mountain height, ad h

dt=

2

2. the

mass being accelerated is less easy to compute exactly, but it is approximately the mass

enclosed in a half cylinder of radius w/2 beneath the mountain (this neglects the mass of the

mountain itself, which is not strictly correct, but if h is small compared to w, the mountain

mass is only a small correction). the mass is then m w L≈π ρ8

2. this yields a simple, second-

order differential equation for the mountain height h as a function of time, t:

d h t

dt

g

wh t

2

2

8( )( ).=

π (B3.1.1)

this equation has the solution

h t h eo

t t( )

/= − collapse

(B3.1.2)

where h0 is the initial height of the mountain and the timescale for collapse is given by:

tw

gcollapse =

π8

.

(B3.1.3)

h

w

m

g

figure B3.1.1 the dimensions and velocity of a linear collapsing mountain of height h and

width w on a strengthless half space of density ρ that is compressed by the surface gravity g on a luid planet. as the mountain collapses vertically it drives a plug of material of mass m

underneath it that lows out through the dashed cylindrical surface.




3.2 Stress and strain: a primer

a full exposition of the continuum theory of stress and strain is beyond the scope of

this book. for the intimate details, the reader is referred to sources such as turcotte and

Schubert’s excellent book Geodynamics (2002). a few simple concepts will sufice for a

general understanding of planetary surface processes, although the actual computation of

stresses under the different loading conditions illustrated later in this chapter requires an

application of the full theory of elasticity.

3.2.1 Strain

Strain is a dimensionless measure of deformation. It is a purely geometric concept that is

meaningful only in the limit where solids are approximated as continuous materials: all

relevant dimensions must be much larger than the atoms of which matter is composed.

Historically, the concept of strain was derived from measurements of the change in length

of a rod that is either stretched or compressed. When a force is applied parallel to a rod of

length l, its length changes by an amount Δl. the length change Δl is observed to be propor-tional to the length l itself, so Δl depends on the size of the specimen being tested. a measure of deformation that is independent of the specimen size is obtained by taking the ratio of

these two quantities to deine a dimensionless longitudinal strain as (see figure 3.1a):

εl

l

l=

∆.

(3.2)

a full description of extensional strain in a three-dimensional body requires three per-

pendicular longitudinal strains, one for each direction in space.

In addition to stretching or compression, a solid can also be deformed by shear, in which

one side of a specimen shifts in a direction parallel to the opposite side. In the special case

x

b

bA

F s

p

l l

Fl

Ac

V

V

(a) (b) (c)

figure 3.1 three varieties of strain. (a) longitudinal strain, in which a block of material of original

length l and basal area Ac is extended an amount Δl by a force Fl. (b) Shear strain, in which the top of a block of height b is sheared a distance Δx relative to its base (to an angle θ) by a differential force Fs. (c) Volume strain, in which a block of original volume V is compressed an amount ΔV by a pressure p.



3.2 Stress and strain: a primer 53

of simple shear the top of a layer of thickness b is displaced by a horizontal distance Δx from the bottom, while its thickness b remains constant. In this case the shear strain is

deined as (figure 3.1b)

ε θs

x

b= ≈

∆

(3.3)

where θ is the slope angle of the sheared material. this angle becomes exactly equal to Δx/b as Δx approaches zero. again, because space is three-dimensional there are three independent shear strains.

Mathematically sophisticated readers may note that the six strains are not vector quan-

tities, but form components of a 3 × 3 symmetric tensor. the three perpendicular lon-

gitudinal strains are the diagonal components and the shear strains are the off-diagonal

components. an important theorem states that the coordinate axes can always be rotated

to a system in which the strain tensor is diagonal. In this coordinate system all strains are

longitudinal, although some may be compressional while others are extensional. a gen-

eral 3 × 3 matrix has 9 components, not 6. the extra three (which form an antisymmetric

tensor) correspond to pure rotations, which, because they do not cause distortions of the

material, are wisely excluded from the deinition of the strain tensor.

finally, if all the dimensions are shrunk or expanded equally, the shape is preserved,

but the volume V changes, and the resulting deformation is described by the volume strain

(figure 3.1c):

εV

V

V=

∆.

(3.4)

there is only one volume strain and it depends entirely on the longitudinal strains,

because it can be expressed as the sum of the three perpendicular longitudinal strains.

3.2.2 Stress

Stress is a measure of the forces that cause deformation. In the limit of small deformations

it is linearly proportional to strain for an elastic material. Just as the strain is expressed as

a ratio of the change in length divided by the length, to make it independent of the size

of the test specimen, stress is expressed as the ratio between the force acting on the spe-

cimen and its cross-sectional area. Deined in this way, stress is independent of the size of

the test specimen and has dimensions of force per unit area, the same as pressure. thus, if

the cross-sectional area of a rod is Ac, and a force Fl is acting to stretch or compress it, the

normal stress in the rod is deined as:

σ ll

c

F

A= .

(3.5)

Similarly to longitudinal strain, there are three normal stresses, one for each perpendicu-

lar direction of space.




Stress is deined as positive when a rod is extended. this makes stress proportional to

strain times a positive number. this is a sensible procedure and is used without further

comment in engineering texts, in which positive stress is tensional. However, in geologic

applications stresses are nearly always compressional. even when stretching does occur, it

is often under conditions of an overall compressional background stress, so that the stress

in the extended direction is simply less compressive than the other directions (in this case,

the stress is often said to be extensional as opposed to tensional). for such applications it

would obviously be simpler if compressional stress is taken as positive. However, such a

convention complicates other simple relations in the full theory of stress and strain. Various

geological authors have tried special deinitions to deal with this problem, although few

have gone so far as to make the constants relating stress and strain negative. turcotte and

Schubert, in their otherwise excellent book, actually switch conventions halfway through,

and other authors recommend changing the sign of the strain deinition. the least drastic

convention, and the one followed in this book, is to deine pressure as the negative of the

average of the three perpendicular stresses, so that compressive (negative) stress always

give rise to positive pressure. this means that a compressional stress acting on a rock mass

is negative.

In close analogy to shear strains, the three shear stresses are deined as the ratio between

a deforming force Fs and, in this case, the basal area of the sheared layer Ab:

σ ss

b

F

A= .

(3.6)

Just as for strains, stresses are components of a 3 × 3 tensor whose diagonal components

are the normal stresses and the off-diagonal components are the shear stresses. (the three

antisymmetric components of the full 3 × 3 tensor are torque densities, which almost never

arise in practice. We do not consider them further.) Stresses are not vectors: the forces are

vectors, but because the forces are divided by an area that also has a direction in space, the

stresses are components of a tensor. Stresses, thus, do not point in some direction in space.

However, it is always possible to rotate the coordinate axes such that the off-diagonal shear

stresses are zero in the new coordinate system, and stresses are sometimes graphically rep-

resented as triplets of arrows of different lengths pointing in perpendicular directions. But

beware! Such arrows cannot be added or subtracted in the same fashion as vectors!

finally, in the special case where the stresses are equal in three perpendicular spatial

directions, the negative of the force per unit area (all directions are equivalent in this case)

is deined as the pressure:

P

F

A= − = −σ vol .

(3.7)

Because stresses, and stress differences in particular, play a major role in determining

the ability of a solid to resist deformation, it is often convenient to single out the three

perpendicular normal stresses in the special coordinate system in which the shear stresses




vanish. these special stresses are called principal stresses and are frequently denoted σ1, σ2, and σ3 for the maximum (most tensional), intermediate, and minimum (most com-pressive) normal stress directions – but be careful of stress conventions here: in geologic

applications the maximum stress is often taken as the most compressive. So long as

this is understood, it causes little dificulty. In the case of hydrostatic stress (pressure)

these principal stresses are all equal. When there are three unequal deviatoric stresses the

deinition of pressure in equation (3.7) is generalized so that p is equal to the negative

average of the three principal stresses. this quantity plays a special role in the tensor

description of stress because it is a rotational invariant, the (negative) trace of the stress

tensor, divided by 3.

Because of the qualitatively different dependence of strength on pressure and shear, the

stress is often separated into a component that depends only on differential stresses, called

the deviatoric stress (often written as σ ′ – thereby forming a test of the readers’ attentive-ness) plus the (negative) pressure. the principal stresses are then written as σ1′-p, σ2′-p and σ3′-p, whereas the shear stresses are the same as before.

the ultimate strength of many materials is often found to depend on the magnitude of the

difference between the maximum and minimum principal stresses, |σ 1 − σ 3|, without any dependence on the intermediate principal stress. a somewhat more complicated measure

of the total distortional stress that does take the intermediate principal stress into account is

called the second stress invariant Σ2 (pressure is the irst invariant):

Σ2 1 3

2

1 2

2

2 3

21

6= −( ) + −( ) + −( ) σ σ σ σ σ σ .

(3.8)

the factor of 1/6 under the square root is a conventional part of the deinition. there is

also a third invariant, whose role in failure mechanics is more complex, and is not consid-

ered further in this text. these quantities are called invariants because their magnitude does

not depend on the orientation of the coordinate system. once their values are established in

one coordinate system, they are the same in all.

It may seem surprising that there is no shear stress term in either of these formulas: after

all, it is common experience that solids break more readily in shear than under compres-

sion. However, shear actually is incorporated, although this may not be apparent. the rea-

son is that shear is one of those off-diagonal components that are intentionally eliminated

by the coordinate rotation that brings the stress tensor to its diagonal form. It can be shown

that a state of pure shear stress σs is equivalent to one in which the coordinate axes are rotated 45° and the principal stresses are σ 1 = −σ 3 = σs.

3.2.3 Stress and strain combined: Hooke’s law

english scientist (and newton’s arch-rival) robert Hooke (1635–1703) recorded some of

the irst observations of the relation between stress and strain in 1665. Working mainly with

springs (Hooke was really interested in clocks) that produce visible deformations under




relatively small loads, Hooke hypothesized a linear relation between longitudinal stress

and strain, now known as Hooke’s law:

σ l = E ε l (3.9)

where the proportionality constant E has dimensions of pressure and is generally known as

young’s modulus, after a much later researcher who studied the extension of elastic rods.

although it was once believed that a single elastic constant is suficient to describe the

stress–strain relation for a given material, it was inally demonstrated in the early 1800s

that at least two constants are necessary to characterize an isotropic solid (in fact, for a

single crystal, up to 21 elastic constants may be necessary, but here we consider only the

minimum required). the second constant is often taken to be the shear modulus μ that relates shear stress to shear strain:

σ s = 2 μ ε s. (3.10)

the factor of 2 is a conventional part of the deinition that derives from the way shear

strain is deined. Because there are two elastic constants they can be, and often are, com-

bined in various ways. for example, pressure and volume strain are related by a constant K

usually known as the bulk modulus:

p = −K ε V (3.11)

(note the minus sign because of the way pressure is deined). Because there are only two

independent stress–strain constants, one of these three must obviously be a function of the

others: It can be shown that E = 9Kμ/(3K + μ).another useful combination is called Poisson’s ratio ν. In figure 3.1a the extended rod

is illustrated as having contracted in the direction perpendicular to its extension. this is a

real, observed effect (indeed, the case of pure extension, without lateral contraction, is very

dificult to realize in practice as it requires tensional loads perpendicular to the extension

axis to maintain a constant cross section). the dimensionless Poisson’s ratio is deined

as the ratio between the amount of lateral contraction and the longitudinal extension of a

laterally unconstrained rod. the deformation illustrated in figure 3.1a actually involves

both a volume change and shear (change of shape), so that the young’s modulus contains

contributions from both the bulk modulus and shear modulus. In terms of Poisson’s ratio,

ν, the young’s modulus is E = 2(1 + ν)μ.relations between stress and strain are generally known as constitutive relations.

Hooke’s law was simply the irst of what is now understood to be a large class of possible

relationships between deformation (strain) and applied force (stress). Such relations may

also involve time: We will shortly meet the concept of viscosity (invented by newton) that

relates the strain rate (the derivative of strain with respect to time) to applied stress. In

modern times the study of the relation between deformation and stress has reached a high

degree of sophistication. this ield is now known under the name of rheology. Because the

materials that make up planets are complex, the rheologic properties of materials as diverse

as rock, air, ice, and lava are crucial for an understanding of how the surfaces of planets and

moons formed and continue to evolve.




the mathematically convenient linear relation between stress and strain does not hold in

all, or even in most, real situations: although stress and strain are always proportional for

suficiently small deformations, when the deformation becomes large enough (and large

may be a strain of only 0.001 – not even visible to the human eye!) the relation becomes

non-linear and catastrophic failure of various kinds may occur (figure 3.2). nevertheless,

the combination of simple constitutive laws, such as that of robert Hooke, and the require-

ment that both internal and external forces are in balance (often known under the name

stress equilibrium) has been immensely fruitful in explaining the ability of planets to sup-

port topographic loads.

3.2.4 Stress, strain, and time: viscosity

Just as ideal elasticity is a useful limit describing the deformation of materials at small

strains, so too is the concept of ideal viscosity. Isaac newton irst recognized viscosity

on the basis of his extensive experimental studies, and proposed an ideal generalization

of his experiments (in fact, newton proposed this property mainly to undermine his rival

Descartes’ vortex theory of planetary motion). Ideal elasticity relates shear stress σs and shear strain εs by a linear equation. Similarly, ideal (or Newtonian) viscosity relates the shear stress and shear strain rate ε ̇ s through a single constant η, the viscosity:

σ s = 2ηε ̇ s. (3.12)

Viscosity has dimensions of stress × time, or Pa-s in SI units. the rules for viscous

low are somewhat more complicated than those of elasticity because the volume strain εV

pla

stic

brittle

ductile

ela

stic

Str

ess

Strain

figure 3.2 In a real solid, stress is linearly proportional to strain only for small stresses and strains

(typically only up to a strain of about 0.001). Beyond this limit the relationship becomes non-linear. In

this regime the low deformation may be reversible (non-linear elasticity) or non-reversible (plastic).

at even larger strains the material may fracture, losing its strength suddenly in a brittle fracture, or

continue to deform to large strains in ductile low.




cannot be a function of time: If it were, the volume of a viscous substance under pressure

would gradually decrease to zero! Discussions of viscous low must, therefore, pay careful

attention to the difference between volume strain and shear strain. In most ideal models the

volume strain is set equal to zero; this is called the incompressible limit. a more realistic,

but mathematically more complex, approximation is to treat the volume strain as elastic

and the shear strain as viscous.

3.3 Linking stress and strain: Jeffreys’ theorem

3.3.1 Elastic deformation and topographic support

the earliest and simplest models of topographic support are derived from applications of the

classic theory of elasticity. this theory combines the full tensor deinitions of stress and strain

with a linear Hooke-type relation between stress and strain (with just two elastic constants, the

minimum number) and the stress equilibrium equations to derive a closed mathematical system.

Within the context of this theory, one can show that, starting from an unstressed initial solid, the

stress and strain throughout the solid are uniquely determined by the forces and displacements

acting on its surface. thus, if we approximate a planet, or some well-deined portion of it, as an

elastic solid, and treat the weight of topography as a load acting on its surface, the stress differ-

ences induced by the topography can be accurately computed throughout its interior.

of course, this is an unrealistically rosy picture of what is actually possible: the

troubles come from the detailed conditions under which elastic theory is valid. Harold

Jeffreys, to whom we owe many of the results that follow, was painfully aware of the

limitations of the elastic model, and he devoted much effort to understanding both its

successes and its failures. the irst dificulty is the obvious limitation of elastic behavior

to small deformations. once failure or low occurs, elastic theory becomes invalid. In

principle this can be addressed by numerical methods and is thus inconvenient but not

insurmountable. the second, more insidious dificulty stems from the condition of an

unstressed initial solid. all planetary surfaces with which we are familiar exhibit a long

history of change, of repeated events that certainly exceeded the limits of linear elasti-

city. So to what extent can the near-surface material be considered initially unstressed?

all planetary materials have mass and all are subject to gravity, so at a minimum, the rocks

beneath the surface must develop suficient stresses to support their own weight. However,

even a liquid, without resistance to deformation (but still resisting volume change!) can

support its own weight. It does this by compressing slightly and thus balancing the gravi-

tational force of the overlying material against the much stronger quantum mechanical

forces that resist the close approach of atoms (gravity eventually wins this struggle in the

stellar collapse to a black hole, but this is far outside the range of planetary processes). the

stresses are hydrostatic in this case, and the pressure p a distance h below the surface of a

body with uniform density ρ and surface gravitational acceleration g is given by:

p = ρgh. (3.13)



3.3 Linking stress and strain 59

although such lithostatic pressures may be very large compared to the stress differences

needed to cause rock failure, the large value of the bulk modulus K for most substances

ensures that the associated volume strain is small. In this case, we can simply add the

lithostatic stress and strain of the subsurface rock to that caused by other loads. this is a

consequence of the linearity of the theory of elasticity: two solutions can always be added

to give a third solution, so long as the boundary conditions of the third solution are the sums

of those of its components.

If the rock beneath a planet’s surface crystallizes from a deep liquid mass, or is heated

to such a high temperature that all differential stresses relax after some time, then the

lithostatic stress state described above can be accurately considered to be the initial state

and the response to any subsequent loads can be computed as elastic additions to this

basic state. unfortunately, most planets are not so cooperative: In most cases one can-

not assume that all differential stresses were erased just before the latest episode of

topographic loading.

another elastic solution useful for describing an initial state is derived from the stresses

that develop in an initially unstressed and very wide elastic sheet that is suddenly subjected

to the force of gravity. the elastic sheet cannot expand laterally; it can only compress ver-

tically. In this case the principal stresses are not all equal (lithostatic), but the vertical stress

σV and horizontal stresses σH differ in magnitude:

σ ρ

σ νν

ρ

V

H

gh

gh

= −

= −−1

(3.14)

where ν is Poisson’s ratio, which can be no larger than 0.5. Poisson’s ratio for most solid rocks is close to 0.25, although it can approach 0.0 for loosely consolidated sediments. In

this solution the magnitude of the horizontal stress is smaller than the magnitude of the ver-

tical stress. the difference between the horizontal stresses and the vertical stress increases

linearly with depth and so, at some large enough depth failure must occur, but this is often

so deep that the solution has great practical value.

alert readers may wonder that this solution has any practical value at all: the idea that

a mass of rock might be assembled in the absence of gravity, which is afterwards magic-

ally turned on, seems so artiicial that it could not apply to any real situation. However, as

demonstrated by Haxby and turcotte (1976), this is precisely the stress state that develops

in a rock mass assembled from the gradual accumulation of a stack of thin, broad and ini-

tially stress-free layers. thus, the stresses that develop in a thick pile of lava lows, or in an

accumulating sedimentary basin, are well described by this model. compilations of verti-

cal and horizontal stress measurements in the earth (McGarr and Gay, 1978) show that, in

many places, such as southern africa or in sedimentary basins in north america, stresses

are bounded between the lithostatic and ininite-layer results (this is not true everywhere:

In canada and much of europe horizontal stresses are much larger than suggested by these

solutions).




although the two basic states just described are frequently useful, they are certainly not

unique: through all six editions of The Earth, Jeffreys invariably emphasized that, due to

the generally unknown history of previous deformation, there are an ininite number of

stress and strain conigurations that are compatible with the presently observed topography.

So why did he devote so much time and effort to obtaining elastic solutions when he did not

believe that such solutions could be accurate? Jeffreys frequently cited a theorem he called

Castigliano’s principle, which asserts: “of all states consistent with given external forces,

the elastic one implies the least strain energy” (Jeffreys, ed. 6, appendix c). thus, to the

extent that the forces acting below a planetary surface tend toward a minimum of energy,

the elastic solution delineates the favored minimum. a second reason is that, although a

given elastic solution may not represent the complete stress state, it does often indicate how

the stresses change in response to a small change in the applied loads. for example, the

formation of a distant impact crater or a change in planetary spin rate or tidal stresses may

cause stress changes that are accurately described by an elastic deformation. In either case,

the elastic solutions are of greater signiicance than the limitations of the strictly conceived

elastic model would suggest.

3.3.2 Elastic stress solutions and a limit theorem

using the full theory of elasticity, stresses can be computed beneath various surface loads,

assuming an initially hydrostatic initial state. contour plots of the second invariant Σ2 for four of these conigurations are shown in figure 3.3a–d. figures 3.3a–c apply to long

loads intended to represent idealized mountain proiles, originally computed by Jeffreys.

figure 3.3d shows the stress differences underneath an axially symmetric idealized impact

crater with a depth/diameter ratio of 0.3.

although the patterns illustrated by these various solutions are diverse in detail, there

are a number of similarities. Most obvious is that the maximum stress differences are not at

the surface, but occur some distance below. thus, most of the weight of a sinusoidal series

of mountain ridges is not supported by the strength of the material in the mountains them-

selves, but by material some distance below. this is an important lesson (one ignored by

the builders of the tower of Pisa): foundations are important! the second important lesson

is that the maximum stress difference is about 1/3 of the total load itself for all four cases

illustrated. these results are summarized in table 3.1, where the depth to the maximum

stress and the maximum stress differences for figures 3.3a–d are listed.

the irst lesson from these solutions, the isolation of the maximum stress region below

the surface, is not strictly valid outside the domain of elastic solutions. More sophisti-

cated analyses, using the theory of plasticity described below, show that, although irst

failure upon loading does, indeed, occur where the elastic solution predicts the max-

imum stress differences, once this failure has occurred the failure zone may work its

way toward the surface, especially if the load has sharp edges, as for a cliff or steep

surface slope. the inal, visible failure may, thus, involve a surface landslide localized at

one of these sharp edges. However, the region over which the strength of the material is




exceeded is far broader than such a surface manifestation and is well delineated by the

elastic solution.

the second lesson from the elastic analysis is more enduring. Generations of struc-

tural engineers have devoted their ingenuity to ways of extending their ability to analyze

the maximum stresses that develop in any given structure. the results of this effort (and

the subject of a huge literature of its own) are the so-called limit theorems. although

theorems of this type do not give the user the detailed distribution of stresses in some

complex structure (this must be done on a case-by-case basis using a full knowledge of

the structure and its history of loading), they do give some overall constraints on how

–3 –2 –1 0 1 2 3

–3

–2

–1

0

1(a) (b)

(c) (d)

+ + + + +

–3 –2 –1 0 1 2 3–3

–2

–1

0

1

–3 –2 –1 0 1 2 3–3

–2

–1

0

1

+

0.0 0.5 1.0 1.5 2.0 2.5 3.0

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

+

0.0

figure 3.3 Stresses below various loads placed on an originally unstressed elastic half space. contours

are of the second invariant Σ2 and are drawn at intervals of 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, and 0.4 of the maximum load. these plots were constructed by summing the fourier components of

the airy stress function that satisies the load boundary conditions. (a) Shows the differential stress

magnitudes beneath a series of very long mountains with sinusoidal hills and valleys. (b) Stresses

beneath a vertical-sided strip mountain. (c) Stresses beneath a long mountain with a triangular proile

and (d) Stresses beneath a circular impact crater with depth/diameter ratio 0.3. Plots are not vertically

exaggerated; horizontal dimensions are in units of the load width. the + sign marks the position of the stress maximum in each plot.




strong materials must be to support some given load, independent of structure and history

of construction.

as summarized by Jeffreys, structural limit theorems assure us that to support a surface

load of order ρgh, somewhere in the body stresses between ½ and 1/3 of this load must be sustained. furthermore, this stress is generally supported at a depth comparable to the load

width (exceptions to this depth rule, such as loads supported by strong, thin plates, usually

imply stresses greatly in excess of the minimum).

this fundamental theorem is so important (and so often overlooked in the planetary lit-

erature!) that I set it out by itself for emphasis:

Jeffreys’ Theorem: The minimum stress difference required to support a surface load

of ρgh is (1/2 to 1/3) ρgh. This stress is usually sustained over a region comparable in

dimensions to the load.

of course, this theorem does not prevent much larger stresses from developing in speciic

situations, but a given topographic load cannot be supported by any smaller stress diffe-

rence. the value of this theorem is that it can be linked to speciic strength models to obtain

quick estimates of the maximum topographic variation to be expected on any given Solar

System body, even when the speciics of interior structure and history are unknown. an

example of this procedure is given in the next section.

3.3.3 A model of planetary topography

consider a generic planetary body (figure 3.4) of mass M, average radius R̅ and average

density ρ̅. the surface acceleration of gravity g is:

g

G M

RG R= − = −

2

4

3π ρ

(3.15)

where G is newton’s gravitational constant.

table 3.1 Elastic stress differences, Poisson’s ratio ν = 0.25

load shape

Maximum stress

difference Σ2/ρghDepth of maximum

below surface

Sinusoidal strip,

wavelength λ0.384 0.289 λ

rectangular strip,

width w

0.352 0.865 w

triangular strip,

basal width w

0.305 0.388 w

axisymmetric crater,

depth/diameter=0.3, diameter D0.359 0.305 D




this relation is exact for a spherical body, and approximate for any other shape. If

the surface has topography of order Δh, and its material is of density ρc, the surface load imposed by this topographic variation is about Δσ = ρcgΔh. applying Jeffreys’ theorem, a minimum stress of magnitude Y must be present somewhere in the body’s interior:

Y G R hc≈ =

1

2

2

3∆ ∆σ π ρ ρ .

(3.16)

rearranging, we obtain an equation that relates the maximum topographic variation, Δh, to some measure of strength, Y.

∆hY

G Rc≈

3

2

1

π ρ ρ.

(3.17)

applying this equation to the earth, take ρ ̅ = 5200 kg/m3, ρc = 2700 kg/m3, R ̅ = 6340 km. We ind:

Δhearth (m) ≈ 80.4 Y (MPa). (3.18)

taking Y ≈ 100 MPa, which is about the crushing strength of granite, we see that the

earth can support abut 8 km of topography – not far off the 8850 m height of Mount

everest or the 11 000 m depth of the Marianas trench, when the buoyancy of submerged

rock is taken into account. However, the dependence of Δh on 1/R ̅ means that, if Y is the

Rminρc g

Rmax

CM

M, ρ

R

∆h

figure 3.4 a simple model of the gravitational forces in an irregular self-gravitating body such as an

asteroid. the average radius is R ̅ and the maximum and minimum radii for points on the surface are

Rmax and Rmin from the center of mass cM. the mean density of the object is ρ ̅.




same for all the terrestrial planets, we should expect 8 km high mountains on Venus, 24

km high mountains on Mars and 50 km high mountains on the Moon. as shown in figures

2.3b and 2.3e, this is not far off for Venus and Mars, but is more than twice the observed

topographic range on the Moon in figure 2.3d. evidently strength is not the major factor

limiting the Moon’s topography: History must play a role, too.

applying this model for topography to the smaller bodies of the Solar System, such as

Phobos, this rock strength limitation leads to ridiculous conclusions about the topographic

ranges on these bodies (see Problem 3.1 at the end of the chapter). one might be tempted

simply to give up and look for factors other than strength that limit topography. However,

as we shall see in the next section, a better appreciation of the concept of strength lets us go

considerably farther down the strength limitation path. In particular, we need to appreciate

the laws that govern the strength of broken rock.

3.4 The nature of strength

3.4.1 Rheology: elastic, viscous, plastic, and more

rheology is the study of the response of materials to applied stress. although stemming

from roots in prehistory, e. c. Bingham (of whom we will learn much more in chapter

5) irst established it as a scientiic discipline in the 1930s. It is not a simple science: real

materials are complex and so is their detailed description. However, much of this com-

plex behavior can be understood in terms of the properties of a number of simple ideal

materials, which are then compounded to approximate real substances. We have already

described ideal elastic and viscous substances. a third ideal behavior is implicit in the idea

of strength: an ideal plastic substance is one which does not undergo any strain at all until

the strength reaches some limiting value, after which the strain increases to any extent con-

sistent with other constraints on the material. of course, no real material behaves in this

way, but many materials do not undergo any very large strains until some limiting stress

is reached, after which strain increases rapidly. a slightly more realistic model is to com-

pound elastic behavior with plastic yielding to arrive at an elastic-plastic substance that

responds to applied stress as an ideal elastic material until the stress exceeds some limit,

after which its strain is limited only by system constraints. then we could add materials

whose elastic strain depends on a non-linear function of stress. We can add time depend-

ence by coupling elastic and viscous behavior. and so on.

this section explores some examples of such compound behavior relevant to understand-

ing planetary topography and its long-term evolution. the irst topic we examine is the

ultimate limits to topographic heights, after which we will look at more realistic limits.

3.4.2 Long-term strength

The ultimate strength of atomic matter. a full understanding of the strength of matter was

achieved only in the mid-twentieth century. Despite the triumphs of quantum mechanics

in explaining the bulk properties of matter in the early twentieth century, an explanation of



3.4 The nature of strength 65

strength came much later. the earliest modern attempt to compute the strength of materi-

als from basic principles was a mitigated disaster: yakov frenkel (1894–1952), in 1926

(frenkel, 1926), constructed a simple model of shear resistance (see Box 3.2 for his deriv-

ation) that relates the ultimate strength, Yultimate, of a material to its shear modulus μ:

Yultimate = μ/2π. (3.19)

Box 3.2 the ultimate strength of solids

the irst estimate of the theoretical upper limit to the strength of a solid was formulated by

yakov (a.k.a. Jacov or James) frenkel (1926). frenkel started from the fact that atoms in a

crystal lattice are uniformly spaced at the interatomic distance a. When a solid is subjected to

shear strain, each plane of atoms parallel to the direction of the strain shifts a small distance

u with respect to the plane immediately above or below. the net shear strain is thus given

by εs = u/a, and is numerically the same at both the atomic and macroscopic scales (see figure B3.2.1). the force resisting this deformation increases as one plane of atoms shifts

over the adjacent plane, because the length of the bonds between each atom and its neighbor

increases. However, when the deformation becomes so large that the atoms of adjacent planes

are midway between lattice sites (that is, at a strain εs equal to ½), the attraction to the next atom in the adjacent plane equals the attraction from the shifting atom’s previous neighbor and

the resistance to deformation drops to zero. further deformation brings each atom into closer

proximity to its new neighbor. new bonds form: the atomic plane snaps into a new position,

jumping forward by one atomic step.

the force between adjacent atomic planes of a strained crystal is thus periodic, with a repeat

distance equal to the interatomic spacing. frenkel assumed that this periodic function would

be the simplest that he could think of: a sine function. He set the force resisting deformation

equal to a constant times sin (2πu/a). Because the maximum value of the sine function is 1 (when u = a/4), the constant equals the ultimate strength of the crystal, Yfrenkel. thus, he supposed that the shear stress is given by:

σ π πεs sYu

aY=

=frenkel frenkelsin sin( ).2

2

(B3.2.1)

to determine the constant, he noted that very small deformations are elastic, and in this

limit σs = μεs. expanding the sine function for very small arguments yields frenkel’s relation for the ultimate strength of a solid in terms of the shear modulus μ,

Yfrenkel =

μπ2

.

(B3.2.2)

although defect-free solids such as ine whiskers and carbon microtubules can approach this

limit, table 3.2 shows that frenkel’s limit greatly overestimates the strength of real materials,

even for rocks at high conining pressures.

accurate computation of the actual strength of materials is not yet possible, so that

measurement and empirical estimates are still necessary to determine the strength of a real

substance under conditions of interest to planetary science.




0

0 0.25 0.50 0.75 1

1

–1

3

2

1

peak

stress

symmetrical

positionnew

position

1 undeformed 2 small strain 3 large deformation --

bonds reform after

plastic flow

Forc

e / Y

ield

Str

ess

Strain, u/a

figure B3.2.1 the theoretical limit to the strength of a solid, based on the model of yakov

frenkel. the graph on the top shows the sinusoidal dependence of shear force on shear strain,

indicating that it is a periodic function of lattice displacement. the lower part of the igure shows

the deformation of a lattice at three different strains, correlated with points on the force–strain

plot above by the circled numbers: (1) is the undeformed solid, (2) has been subjected to a small

strain, while (3) indicates a strain so large that the atoms in the solid are again in register with

their neighbors, so that the shear force vanishes.

Box 3.2 (cont.)

the shear modulus has been measured for a large variety of materials. It is a bulk

property that can now be computed from irst principles for many single crystals.

although frenkel’s formula is elegantly simple, it is also grossly inadequate: as shown

in table 3.2, the actual measured strength of most materials is a factor of 100 or more

smaller than the frenkel limit. nevertheless, the frenkel limit is not wholly wrong or

useless: the strength of a few materials, such as carefully prepared single crystals or

ine carbon ibers, does approach this limit. However, the frenkel limit clearly does

not capture the factors controlling the strength of the materials we are likely to meet in

planetary interiors.

the principal shortcoming of frenkel’s strength estimate is its neglect of defects. rocks

are composed of crystals of individual minerals. While the crystals themselves might be

strong, they are bonded through weaker surface interactions. Most igneous rocks, such as

granite or basalt, have cooled through a large range of temperatures and, because of the

different thermal expansion coeficients of their constituent minerals, tiny grain-boundary

cracks develop in abundance. Sedimentary and metamorphic rocks also contain vast num-

bers of microscopic cracks and weak bonds between individual grains. all rocks contain




macroscopic cracks in the form of joints. In addition to cracks between mineral grains, the

minerals themselves inevitably contain arrays of a peculiar sort of strength-related line

defect called dislocations. first described in the 1950s by engineers studying the creep

elongation of turbine blades in high-temperature jet engines, dislocations low under

stresses far below the frenkel limit. It is only by studying the properties and interactions of

entities such as cracks and dislocations that progress has been made in understanding the

practical limitations on the strength of materials.

although the strength of materials is a large ield of endeavor in itself, one too vast to

cover in this book (references for this literature are provided at the end of this chapter), the

basic take-away lesson is that defects rule the macroscopic strength properties of materials.

one cannot expect planetary materials to be stronger than a small fraction of the frenkel

limit. and, in spite of a half-century of progress in understanding the fundamental basis of

strength, there are so many complex contributing factors that the strength of a particular

material under given conditions of pressure, temperature, and chemical environment is still

best determined by experiment.

traditional material science focuses on the strength properties of metals. only recently

have the much more complex problems presented by the strength of ceramics and geologic

materials, such as rocks, become amenable to rational explanation. naturally, experiment-

ers did not wait for theoreticians to make up models of the strength of rock, so that much

of our present understanding is based upon empirical observations.

Built upon sand: The strength of broken rock. Most experts on asteroids now believe that

all but the very smallest asteroids (bigger than a few tens of meters in diameter) are better

described as fragmented rubble piles than as solid chunks of rock. unlike solid rock, rubble

table 3.2 Theoretical vs. observed material strength

Solid material

yultimate= μ /2π (GPa)a

yobservedat p = 1 and 5 (GPa)b

Iron, fe 13.0 0.11–1.0

aluminum, al 4.14 0.10–0.30

corundum, al2o3 25.9 0.26–0.92

Periclase, Mgo 20.9 0.14–1.07

Quartz (opal), Sio2 7.08 0.35–1.8

forsterite, Mg2Sio4 12.9 1.13 (p = 0.5 GPa)c

calcite, caco3 5.09 0.27–0.84

Halite, nacl 2.34 0.09–0.29

Ice, H2o 0.54 0.20–1.0d

a elastic moduli from Bass (1995).b at 23°c from Handin (1966) table 11–9, except as noted.c at 24°c Handin (1966), table 11–3, Dun Mtn., nZ, peridotite.d at 77–115 K; extrapolated from Beeman et al. (1988).




piles have no tensile strength. their entire ability to resist changes in shape depends on the

frictional forces acting across the rock–rock contacts between their components.

coulomb in 1785 irst formulated the laws governing the mechanical behavior of a mass

of broken rock (or a pile of sand). Because the frictional resistance at a rock–rock con-

tact is proportional to the force pushing the rocks together, the strength of a mass of bro-

ken rock is proportional to the pressure. this fact was irst clearly stated by leonardo da

Vinci (1452–1519) in the ifteenth century, but not published by him. Guillaume amontons

(1663–1705) in 1699 resurrected this relation from da Vinci’s codices. this behavior is in

stark contrast to the strength of ductile metals, such as aluminum or steel, which is nearly

independent of pressure. Many experimental studies of the strength of sand or soil show

that the mass begins to yield when the applied shear stress σs reaches a constant fraction of the overburden pressure p:

|σ s| = ff p = tan φ f p (3.20)

where ff is the coeficient of friction and φ f is the related angle of internal friction. this angle is also closely related to φ r, the angle of repose, which is the maximum steepness of a slope composed of this material (See Section 8.2.1 and table 8.1 for more on internal fric-

tion). this coeficient is typically about 0.6 for most geologic materials (including water

ice well below its freezing point), making φ f about 30°.applying this formula to a model of small-body topographic support, the most obvious

evidence of topography on small bodies is the difference between their longest and shortest

dimensions, Rmax − Rmin (refer back to figure 3.4) this out of roundness corresponds to a

load of breadth comparable to the mean radius of the body itself, R ̅. the stress support-

ing this load is, thus, localized deep within the body. the average pressure in the center

of a homogeneous body (ρc = ρ̅) is p g Rctr =1

2ρ , so that the strength, Y, or resistance to

yield, is Y ≈ ff pctr. Inserting this into the equation for Δh, we ind that a small-body model of strength implies:

Δhsmallbody ≈ ff R̅. (3.21)

another way of deriving the same result is to note that a constant coeficient of friction

implies a constant angle of repose, which is nearly equal to the angle of internal friction.

Imagine a hypothetical, maximally out-of-round asteroid constructed in such a way that

every slope on its surface is at the angle of repose in its local gravitational ield (such a

shape has now been constructed by Minton, 2008). although the precise shape is complex,

it is clear that, in traversing the surface of the asteroid from equator to pole, a distance of

(π/2)R ̅, up (or down) a constant slope of angle φr, an elevation change of the order of (π/2)R̅ tan φr must take place. this yields essentially the same Δhsmallbody as above.

this small-body topography model predicts that the maximum fractional deviation from

sphericity, (Rmax − Rmin)/R̅, is actually independent of size. this is in strong contrast to the

constant-strength model derived for the earth, which suggests that, as a body becomes lar-

ger, its shape becomes relatively closer to a spheroid because ( ) / ∝ /max minR R R R ,− 12

so

that the ratio decreases as R ̅ increases.




How do these model predictions fare against reality? figure 3.5 plots the maximum frac-

tional deviation from sphericity against mean radius for a variety of Solar System objects.

It is clear that the topography of the smaller bodies does, indeed, follow a law that suggests

the dominance of frictional strength. there is no obvious tendency for the fractional topo-

graphic deviation to decrease with increasing size. However, at a radius of about 200 km

the frictional relationship breaks off and the maximum topographic deviations of the larger

planets and moons decrease sharply with increasing diameter, following an approximate

1/R2̅ dependence on the log–log plot. for these large objects greater size does imply greater

smoothness. the trend of the curve for larger planetary objects suggests that the ultimate

strength of planetary crusts is about 0.1 GPa.

the constancy of the maximum fractional deviation for small objects is a direct con-

sequence of the ability of pressure to increase the strength of broken rock materials.

obviously, however, this frictional increase in strength has its limits. this fact is also clear

from laboratory measurements of rock strength: as shown in figure 3.6, the frictional

regime holds up to some maximum stress, generally a few GPa, when the intrinsic strength

of the rock is reached and yielding occurs in spite of increasing overburden pressures. as

in the large–planet topography model, it seems that the ultimate limit to topography lies

in the ultimate ability of matter to resist deformation. It is thus worth inquiring just what

determines this resistance.

David Griggs and the strength of rocks. the most obvious feature of the rocks outcrop-

ping on the surface of the earth is that they are pervaded by fractures at all scales. How

these fractures actually form, however, is much less obvious. It took many years before

experimenters could reproduce the pressures and temperatures existing in the earth’s

0.001

0.01

0.1

1

10

1 10 100 1000 10000

(Max

-Min

)/M

ean R

adiu

s

Mean Radius, km

10 MPa

0.1 GPa

1 GPa

= 30°

figure 3.5 the ratio of the maximum elevation difference to the radius for various Solar System

bodies as a function of diameter. up to a diameter of about 200 km, this ratio is nearly constant, as

expected for rubble piles supported only by frictional strength. above this diameter the ratio falls off,

consistent with an ultimate planetary crustal strength of about 0.1 GPa. the solid dots are silicate

bodies and the open circles are icy. the data suggests that icy bodies are weaker than silicate objects

although they have similar friction coeficients.




interior and come to an understanding of how rocks break. Indeed, this is still an active area

of research in the earth sciences. David Griggs (1911–1974) was one of the irst people to

systematically investigate rock fracturing under high pressures and temperatures. Griggs’

interest in geologic processes began as a boy, when he accompanied his father, geologist

robert Griggs, on a national Geographic expedition to study the deposits of the fam-

ous 1912 eruption of Mount Katmai in alaska (Griggs, 1922). from his experience in

the ield, David decided to study how rocks break deep within the earth. He sought out

Percy Bridgeman at Harvard university and signed on as his graduate student in 1933.

Bridgeman’s laboratory was one of the few places in the world where pressures approach-

ing those deep in the earth’s crust could be attained.

Griggs eventually perfected an apparatus widely known as a Griggs’rig that could both

compress and heat a small rock sample, typically a cylinder a few centimeters in length and

diameter, while subjecting it to controlled differential stresses. continuing his work after

World War II at ucla, and accompanied by a growing number of similarly motivated

experimenters, he showed that, unlike metals, the fracture strength of rock is a strong func-

tion of both pressure and temperature.

It has long been known that metals and alloys, such as iron or steel, fail at similar stresses

under both compression and tension. Ideal plasticity is a useful approximation to metal

failure, in which half the stress difference at failure (equivalent to the shear stress through

a coordinate rotation) is assumed to be a constant Y, the yield stress:

σ

σ σs Y=

−=1 3

2.

(3.22)

0

25

50

75

–50 0 50 100 150

Shea

r S

tres

s s, M

Pa

Pressure p, MPa

YM

= 100 MPa

Y0 = 20 MPa

fF = 0.6

Intact rock

Fractured rock

Tension Compression

Y0

–Y0 /2

figure 3.6 yield stress of a typical intact rock specimen (heavy line) described by the lundborg

strength envelope, equation (3.23). note the substantial tensional strength (equal to Y0 / 2 by the

Brace construction, which is, nevertheless, weaker than the extrapolation of the lundborg strength

envelope, shown by the dotted line, would suggest) indicated on the negative pressure axis. Shown

also as a heavy dashed line is the yield curve for a fractured rock specimen for which the shear

resistance is entirely due to friction.




the yield stress of metals is, to a good approximation, independent of pressure and

strain, although it declines with increasing temperature. Because of its utility in engin-

eering, the theory of failure of ideally plastic materials is highly developed, in spite of

serious mathematical dificulties that stem from this very lack of dependence on strain

(Hill, 1950).

experimental studies of rock fracture show, however, that the strength of rock depends

very strongly on pressure, at least up to pressures approaching 5 GPa (50 kilobars).

Many analytic representations of the failure strength of rock have been proposed; among

them, one that seems to it many materials was suggested by lundborg (1968) for

unfractured rock:

σ sf

f

M

Yf p

f p

Y Y

= ++

−

0

0

1

(3.23)

where Y0 is the strength at zero pressure, often called cohesion, and YM is known as the von

Mises plastic limit of the material. YM limits the maximum stress that can be achieved at

arbitrarily high pressure. the lundborg form of the failure law is illustrated in figure 3.6

and some representative values of the parameters are listed in table 3.3.

although the lundborg law, and others like it, gives a good description of the failure of

rock over the full range of pressures from very low to very high, much more data has been

collected in thelow pressure regime where a linear version is generally adequate. thus, when p


the sloping, low-pressure portion of the failure law illustrated in figure 3.6 is superi-

cially similar to that of sand. However, in this case the pressure coeficient ff is less obvi-

ously related to friction, although it is often referred to as a coeficient of internal friction,

presumably because it is dimensionless and relates strength linearly to overburden pressure,

as does the true friction coeficient. numerically, it is also similar to the coeficient of rock-

on-rock friction, although the reader should not confuse the two: ff is the (approximate)

linear slope of the strength envelope that deines the stress conditions under which intact

rock fails, whereas fB is the (static or starting) coeficient of friction of a pre-existing planar

rock fracture sliding over another. the difference between these two curves is responsible

for the brittle–ductile transition that gives rise to discrete faults in rock, as will be discussed

in more detail in Section 4.6.1.

extensive tables of the strength envelopes of rocks under various conditions can be found

in Handin (1966) and lockner (1995). the ultimate strength limit of about 0.1 up to 1 GPa

for real rocks is in fair agreement with the observed trend of topographic deviations on the

larger planets illustrated in figure 3.5. It, thus, appears that we presently have a good irst-

order understanding of the strength properties of planetary bodies, although many details

remain to be worked out.

the presence of pre-existing fractures in most large rock masses greatly complicates ana-

lyses of the strength of rock. the actual strength of a large volume of rock generally lies some-

where between that of intact rock and that deined by the coeficient of friction (the dashed

line in figure 3.6). a constant value of the friction on a pre-existing fracture, fB 0.85 (up to a mean pressure p of about 100 MPa; the slope is somewhat less at larger pressure) is often

known as Byerlee’s law after the researcher who showed that this value describes the friction

of a wide variety of rock surfaces (Byerlee, 1978). In its exact form Byerlee’s law states:

σσ σ

σ σsn n

n n

=<

+ ≥0 85 200

50 0 6 200

.

.

MPa

MPa

(3.25)

where σn is the normal stress across a fracture, σs is the shear stress and all stresses are in megapascals.

table 3.4 Low-pressure failure envelope for representative rocks

rock friction coeficient, ff cohesion, Y0(MPa)

Westerly granite @ 500°c 0.6 50

Pennant sandstone @ 25°c 0.97 35

limestone @ 25°c 0.75–1.6 3.5–35

Siltstone @ 25°c 0.55 21

chalk @ 25°c 0.38 0.9

Data from Handin (1966).




note that the mean pressure, p, in equation (3.23) is somewhat confusingly equal to the

negative of either one-half of the sum of the maximum and minimum principal stresses,

or (more correctly, if less frequently seen) to one-third of the sum of all three principal

stresses. a similar equation is often written in which, in the location occupied by the term

p in equation (3.23), a term for the normal stress acting across the failure plane appears

instead. Byerlee’s law is strictly valid only for this normal stress. the disadvantage of this

formulation is that the failure plane must be known before the equation can be applied.

thus, for the present goal of deining a strength envelope, a formulation in terms of stress

invariants (pressure and shear stress) is preferable. the wary user of data tables is careful

to make sure which deinitions are in use before accepting a given coeficient of internal

friction at face value!

the mean pressure, p, in the equation (3.23) must be modiied by subtracting the pore

luid pressure, p → p − pf, when the rock is pervaded by a luid that itself is at some hydro-static pressure pf. this modiication is very important when a luid such as water or oil on

earth, or methane on titan, is present. It was irst introduced by terzaghi (1943) for soils,

and by Hubbert and rubey (1959) for rocks. Its detailed implications are the subject of a

large literature. It will be discussed further in Section 8.2.1, but sufice it to say now that

high luid-pore pressures cause substantial weakening of rock through this pressure sub-

traction effect.

the coeficient Y0 in equation (3.23) is the zero-pressure strength or cohesion.

Mathematically, it is the intercept of the strength envelope with the zero-pressure axis (see

figure 3.6). Physically, it represents the adhesion of crystals in the rock to one another and

can range from only a few megapascals for weak sedimentary rocks to several tenths of a

gigapascal for intact granite. It is strongly affected by pre-existing cracks in the rock and

drops to zero in a fully fractured rock mass. an extrapolation of this line to negative values

of p intercepts the pressure axis (zero shear stress) at pT = −Y0 /ff. this intercept corresponds to the tensile strength of the rock. the linear extrapolation yields an overestimate of the

actual yield stress by a factor of two to three: More sophisticated models based on crack

theory (Brace, 1960) give a different, and more accurate, analytic form for tensile stresses

that is indicated by the heavy yield curve on figure 3.6.

the slope of the failure curve decreases at large values of the average pressure, and the

maximum shear stress that the rock can sustain approaches a constant YM, independent

of pressure. this rollover occurs when the frictional stress of sliding on inter- and intra-

crystalline cracks approaches the intrinsic strength of the individual crystals. a full under-

standing of this process is still under development, but the general outlines are now in fairly

good agreement with observations (ashby and Sammis, 1990). this change in the depend-

ence of the strength on pressure is known as the brittle–ductile transition, for reasons that

will be discussed in more detail in the next chapter, Section 4.6.1. It occurs at, or near, the

point where the failure curve for fractured rock crosses that for intact rock in figure 3.6.

the ultimate yield stress YM in equation (3.23) is, as shown in figure 3.6, still far

below the frenkel limit because of intra-crystalline defects such as dislocations. although

independent of pressure, by deinition, it does depend strongly on temperature. there is




no universal law for this temperature dependence, which must be determined empirically,

but it is clear that the strength must vanish at the melting temperature, Tm. using this hint,

a widely used approximation to the temperature dependence is to multiply both Y0 and YM

by the same factor:

FT T

TT

m

m

=−

2

(3.26)

which assures that the strength falls to zero as the temperature approaches the melting

point. the exponent in this relation is purely empirical, chosen to it a large body of data

on both metals and rocks.

3.4.3 Creep: strength cannot endure

David Griggs and the low of rocks. When David Griggs began his now-classic work in

1933 he was already the veteran of many geologic ield excursions and knew from personal

experience that the rocks of the earth’s crust often show signs of large amounts of deform-

ation without fracture. this luid-like deformation had long been attributed to the high

pressure and temperature within the inaccessible depths of the earth, but no one understood

the rates or conditions under which this low occurred. Griggs began his lifework with a

relatively simple apparatus that measured the slow deformation of rocks under an applied

load as a function of time, initially working at room temperature and pressure (figure 3.7).

although he found that most rocks deform elastically only for periods of time less than a

year, he discovered a few that exhibited slow pseudoviscous low or creep according to a

simple law relating the strain ε and time t:

ε = A + B log t + C t (3.27)

where the constant A represents instantaneous elastic deformation, B a kind of decelerating

creep now often called primary creep, and C is the rate of steady, long-term low. although

the primary creep term is important for short-term low processes, such as the response to

luctuating tidal stresses or the small strains that accompany planetary reorientation and

spin changes, most geologic interest centers on the third, steady-state term, because it rep-

resents deformation that increases steadily with increasing time, apparently without limit.

In this respect the low of rocks resembles that of more familiar viscous liquids, such as

honey, motor oil or tar.

Sixty years of subsequent research by Griggs and a large cadre of laboratory geologists

who recognized the importance of this research has shown that the rate of steady-state

creep is a function of stress, temperature, and pressure, as well as rock composition, grain

size, presence or absence of water, trace elements, and a host of other factors. Most creep

experiments can be it by a formula of the form:

C A ec

n

Q

RT= =−ε σsteady

*

(3.28)




where Ac is a constant with dimensions (stress)–n time–1, σ is deviatoric stress, n a dimen-

sionless constant, Q* is activation enthalpy (this term incorporates most of the pressure

dependence because Q* = E* +pV*, where p is pressure and E* and V* are constants), R the gas constant, and T is absolute temperature. the dot over the strain ε, following newton’s luxion notation, indicates differentiation with respect to time.

It is often convenient to express the rate of steady-state creep, equation (3.28), in terms

of an effective viscosity, even though it depends on the stress level. adapting the deinition

of viscosity, equation (3.12), the effective viscosity ηeff is deined as:

ησε σeff steady

= = −s

Q

RT

c

n

e

A2 2 1

*

.

(3.29)

test specimen

weight

strainindicator

l

Str

ain

Time

steady creep

0

fracture

(b)

(a)

figure 3.7 Schematic representation of a creep experiment on rock, similar to Griggs’ 1933 room-

temperature measurements. (a) the test specimen, of original length l, is mechanically loaded (by a

weight and a lever) while its delection is measured on a sensitive scale. (b) Schematic creep curve,

showing strain as a function of time after loading. the curve shows three distinct portions after the

initial elastic delection: a period of decelerating creep, a long period of steady creep and, for lab

specimens, a inal acceleration just before rupture.




this deinition of viscosity generalizes newton’s original deinition, which applies to the

case n = 1. It has now become common to refer to the case n = 1 as “newtonian viscosity” and to use the term “viscosity” in the broader sense for any value of n, as long as it refers

to a low law in which the strain rate is a function of stress.

unlike viscous liquids, the power n relating stress and strain rate is usually larger than

1 for creeping rocks and minerals, justifying the use of the term “pseudoviscous” for this

kind of low. Doubling the stress on materials such as ice or olivine may cause the creep

rate to increase by a factor of 10, in strong contrast to ideally viscous materials in which

the creep rate only doubles. It is also important to realize that creep rate depends exponen-

tially on the temperature. although rocks deform very slowly at low temperatures, as the

temperature climbs toward the melting point the creep rate increases rapidly (by as much

as a factor of 10 for each 100°c increase in temperature for many rocks). a useful approxi-

mation is that for most materials, creep rates become important over geologic time periods

(millions of years, which implies ε ̇ steady ≈ 10−13 s−1 or less) when the temperature reaches one-half the melting temperature, T ~ 1/2Tm. a useful simpliication of the temperature

dependence of the creep rate is to absorb the activation energy and melting temperature

into a constant g and express the temperature as the dimensionless ratio T/Tm, the homolo-

gous temperature:

C A ec

ng

T

T

m

= =−ε σsteady . (3.30)

table 3.5 gives typical values for Ac, n, Q*, Tm, and g for a few materials of geologic and

planetary interest.

extensive tables, such as that of Kirby and Kronenberg (1987a, b) and evans and

Kohlstedt (1995), have been compiled to categorize the creep of rocks, and theoretical

models have been developed to explain this low behavior in terms of diffusion and dis-

location motion (e.g. evans and Kohlstedt, 1995; Poirier, 1985). However, for the purposes

of this book the principal concept to remember is that at high temperatures rocks can low

like liquids over geologic timescales.

J. C.Maxwell and the viscosity of “elastic solids.” observation and experiment have

taught us that cool materials (that is, materials at temperatures well below their melting

point) deform elastically under applied loads, while hot materials gradually low. elastic

behavior is mostly recoverable: that is, when the load is

Documents

Strength versus gravity - UChicago GeoScigeosci.uchicago.edu/~kite/doc/ch3_of_melosh.pdf52 Strength versus gravity 3.2 Stress and strain: a primer a full exposition of the continuum